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Beliebt Rechnen >

integral of (4x-5)/((x-1)(x^2+1))

Lösung

\int \frac{4 x - 5}{( x - 1 ) ( x ^{2} + 1 )} d x

Lösung

- \frac{1}{2} ln ∣ x - 1 ∣ + \frac{1}{4} ln x^{2} + 1 + \frac{9}{2} arctan (x) + C

Schritte zur Lösung

\int \frac{4 x - 5}{( x - 1 ) ( x ^{2} + 1 )} d x

Multipliziere aus

\frac{4 x - 5}{( x - 1 ) ( x ^{2} + 1 )} : \frac{4 x}{( x - 1 ) ( x ^{2} + 1 )} - \frac{5}{( x - 1 ) ( x ^{2} + 1 )}

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{4 x}{( x - 1 ) ( x ^{2} + 1 )} d x - \int \frac{5}{( x - 1 ) ( x ^{2} + 1 )} d x

\int \frac{4 x}{( x - 1 ) ( x ^{2} + 1 )} d x = 4 (\frac{1}{2} ln ∣ x - 1 ∣ + \frac{1}{2} (- \frac{1}{2} ln x^{2} + 1 + arctan (x)))

\int \frac{5}{( x - 1 ) ( x ^{2} + 1 )} d x = 5 (\frac{1}{2} ln ∣ x - 1 ∣ + \frac{1}{2} (- \frac{1}{2} ln x^{2} + 1 - arctan (x)))

= 4 (\frac{1}{2} ln ∣ x - 1 ∣ + \frac{1}{2} (- \frac{1}{2} ln x^{2} + 1 + arctan (x))) - 5 (\frac{1}{2} ln ∣ x - 1 ∣ + \frac{1}{2} (- \frac{1}{2} ln x^{2} + 1 - arctan (x)))

Vereinfache

4 (\frac{1}{2} ln ∣ x - 1 ∣ + \frac{1}{2} (- \frac{1}{2} ln x^{2} + 1 + arctan (x))) - 5 (\frac{1}{2} ln ∣ x - 1 ∣ + \frac{1}{2} (- \frac{1}{2} ln x^{2} + 1 - arctan (x))) : - \frac{1}{2} ln ∣ x - 1 ∣ + \frac{1}{4} ln x^{2} + 1 + \frac{9}{2} arctan (x)

= - \frac{1}{2} ln ∣ x - 1 ∣ + \frac{1}{4} ln x^{2} + 1 + \frac{9}{2} arctan (x)

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{2} ln ∣ x - 1 ∣ + \frac{1}{4} ln x^{2} + 1 + \frac{9}{2} arctan (x) + C

Graph

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